Time-reversal symmetry breaking phase in the Hubbard model: a VCA study
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Time-reversal symmetry breaking phase in the Hubbard model: a VCA study
Xiancong Lu,
1, 2
Liviu Chioncel,
3, 4 and Enrico Arrigoni Department of Physics and Institute of Theoretical Physics and Astrophysics, Xiamen University, Xiamen 361005, China Institute of Theoretical and Computational Physics,Graz University of Technology, A-8010 Graz, Austria Augsburg Center for Innovative Technologies, University of Augsburg, D-86135 Augsburg, Germany Theoretical Physics III, Center for Electronic Correlations and Magnetism,Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany
We study the stability of the time-reversal symmetry breaking staggered flux phase of a single bandHubbard model, within the Variational Cluster Approach (VCA). For intermediate and small valuesof the interaction U , we find metastable solutions for the staggered flux phase, with a maximumcurrent per bond at U ≈ .
2. However, allowing for antiferromagnetic and superconducting long-range order it turns out that in the region at and close to half filling the antiferromagnetic phaseis the most favorable energetically. The effect of nearest-neighbour interaction is also considered.Our results show that a negative nearest-neighbour interaction and finite doping favors the stabilityof the staggered-flux phase. We also present preliminary results for the three-band Hubbard modelobtained with a restricted set of variational parameters. For this case, no spontaneous time-reversalsymmetry breaking phase is found in our calculations.
PACS numbers: 71.10.Fd, 71.27.+a, 71.10.Hf, 74.20.-z
I. INTRODUCTION
The Hubbard Hamiltonian is perhaps the simplest pos-sible model to describe electronic interactions in solids.It is however, difficult to solve and even after severaldecades of research there are still many open questionsabout its basic features. In particular many competingphases have been proposed as candidates for the groundstate in special parameter regions. The staggered fluxphase [1, 2], known also as ”orbital antiferromagnet” [3],was proposed to describe the ground state of Heisen-berg Antiferromagnetism. Many properties of the stag-gered flux phase have been discussed in the past. How-ever, less numerical evidence exists that support the ex-istence of this phase in the Hubbard or t-J model [4–10]. The phenomena of pseudogap in cuprate supercon-ductor contributed to the increased interest in the time-reversal symmetry broken phase. Chakravarty et al. [11]argued that the pseudogap region is due to the compe-tition between d -wave superconductivity and d -densitywave (DDW) state [12, 13], which actually is the stag-gered flux state breaking both time-reversal and trans-lational symmetry. Based on the mean-field analysis ofthree-band Hubbard model, Varma showed that a time-reversal symmetry breaking phase, the “circulating cur-rent phase” [14–17], is stable in some parameter regions.The orbital current of this phase is circulating along theO-Cu-O plaqette in each cell and thus, in contrast toDDW, translational symmetry is preserved. Motivatedby this proposal, several theoretical and experimentalgroups have tried to find the signatures of orbital cur-rent in the CuO planes [18–21]. However, no agreementshave been obtained so far. Recent polarized neutrondiffraction experiment shows that the spontaneous cur-rent occurs in loops involving apical oxygen orbitals [22].This picture is consistent with Variational Monte Carlo (VMC) calculations [23], although the authors observethat the computed current of three-band Hubbard modelis smaller, for larger system sizes, therefore computationson larger system are required.There have been a lot of efforts to search for the stag-gered flux phase in the two-leg ladder, which is easierto study and can shed some light on the full two dimen-sion lattices. By using the highly accurate density-matrixrenormalization group technique, Marston et. al. [24]and Schollw¨ock et al. [25] found the evidences for theexistence of staggered flux phase at and away from halffilling, respectively. It has been shown that complex in-teractions are needed to stabilize this phase [24, 25]. Ana-lytical studies using bosonization/renormalization group(RG) method also found stable regions of this phasefor weak interactions [26–29]. In the present work weadopt the Variational Cluster Approach (VCA) to studythe time-reversal symmetry breaking phase. VCA allowsto evaluate the single-particle [30] and two-particle [31]spectral functions, and, to include symmetry breakingphases [30, 32, 33], such as Antiferromagnetism and Su-perconductivity, therefore it can be easily extended toinclude time-reversal symmetry breaking phase.The paper is organized as follows. In Sec. II, wepresent the Hubbard Hamiltonian in a form that includesthe single particle parameters that are relevant for thetime-reversal symmetry breaking. We review briefly theVCA method and introduce the coupling fields associ-ated with the time-reversal symmetry breaking, so called”Weiss” fields. These are the hopping ∆ t and its phase∆ φ , together with the current per bond which arises asa natural order parameter. The search for the sponta-neous time-reversal breaking state within the interact-ing Hubbard model is presented in Sec. III. The follow-ing sections Secs. III A and III B discuss the stabilityof staggered flux phase at and away from half-filling forthe single band Hubbard model. Our results indicatethat the stable ground state for the interacting Hubbardmodel at half filling is the antiferromagnetic state. Awayfrom half-filling, the competition with the superconduct-ing phase is also investigated. Sec. III C discusses theeffect of nearest-neighbour interaction on the staggered-flux phase. In Sec. III D, we extend the procedure de-scribed in Sec. II to the three band Hubbard model andsearch for a spontaneous time-reversal symmetry break-ing phase in this model. Finally, we draw our conclusionsin Sec. IV. II. HAMILTONIAN AND VARIATIONALCLUSTER APPROACH
We consider the following 2D Hubbard model on asquare lattice: H = − X
FIG. 2: (Color online). Difference between the grand poten-tials computed with variational parameters, such as the hop-ping ∆ t , phase factor ∆ φ , and staggered magnetic field h AF ,with respect to the grand potential Ω(0) when no variationalparameters are considered. The calculations are performed athalf filling. In this section, we study the stability of the sponta-neous staggered flux phase in the Hubbard model with real hopping matrix (i.e., φ ij = 0 in Eq. (1)). The situa-tion is different from what we discuss in Section II wherethe electrons are under an external staggered magnetic field. In order to allow for the time-reversal symmetry-breaking order, we introduce a Weiss field H ′ T R (cf. Eq.(3)) in the reference system. For the staggered flux phase,the absolute value of the phase | ∆ φ ij | is chosen to beequal for all bonds (i.e, ∆ φ ) and the staggering configu-ration is shown together with the VCA reference clusterin Fig. 1. We stress that it is important to choose areference cluster with even number of plaquettes, so thatthe net “magnetic field” in the cluster is zero. A. Staggered flux phase in the Hubbard model athalf-filling
In Fig. 2 we plot the difference between the valuesof the grand potential obtained when variational param-eters are included relative to the cases when no varia-tional parameters are considered. The results presentedare for the case of half-filling and different values of U.In the small U region (i.e., from 0 to 3.6), the valuesof the grand potential Ω(∆ t, ∆ φ ) with both ∆ t and ∆ φ taken as variational parameters are always smaller thanthe grand potential Ω(∆ t ) with a single variational pa-rameter ∆ t , the latter being also smaller than the grandpotential obtained in a calculation with no variationalparameters Ω(0). Accordingly, the staggered flux phaseis lower in energy than the correlated paramagnetic (non-magnetic) state of the tight-binding Hamiltonian at half-filling. At U ≈ . φ vanishes indicatinga phase transition to a phase without time-reversal sym-metry breaking. This suggests that the staggered fluxphase is favored by small values of U . This situation wasnot considered within Ref. [1], which predicts that thestaggered flux phase is stable at large U limit, a conse-quence of the mean-field approach. At half filling oneshould expect that the ordered Antiferromagnetic phasealso plays an important role in the Hubbard model. In or-der to compare the stability of the flux phase with respectto the Antiferromagnetic long-range order we included aNeel Antiferromagnetic Weiss field H ′ AF = h AF X i ( n i ↑ − n i ↓ ) e i QR i (5)into the reference system. Here, h AF is the strength ofa staggered magnetic field and Q = ( π, π ) is the Anti-ferromagnetic wave vector. The calculations within thelong-range ordered state shows that Ω(∆ t, ∆ φ ) is largerthan Ω( h AF ) for values of U > .
9, and is lower thanΩ( h AF ) when U < .
9. But for a complete comparison,Ω(∆ t, ∆ φ ) has to be compared with Ω(∆ t, h AF ) whenboth ∆ t and h AF are treated as variational parametersin the long-range ordered state. As shown in Fig. 2, inthis case Ω(∆ t, ∆ φ ) is always larger than Ω(∆ t, h AF ) forthe whole range of U .Although our results indicate that the Antiferromag-netic state is more stable than the staggered flux phasefor the half-filled 2D Hubbard model, the latter can bestill considered as a metastable phase whose fluctuationsaffect the physics of the system, or which may becomestable whenever Antiferromagnetic long-range order issuppressed by some other mechanism.In order to study the metastable staggered flux statein more detail, we plot in Fig. 3a the difference in grandpotentials Ω(∆ t, ∆ φ ) − Ω(∆ t ) (symbol – (cid:3) , with label-ing on the left) and the corresponding current J as afunction of interaction U at half filling. The differencebetween the values of the grand potential increases withU, reaches a maximum for the value of U ≈ .
2, and thendecreases to zero for U ≈ .
6. The largest difference be-tween Ω(∆ t, ∆ φ ) and Ω(∆ t ) is 0 . t ≈ meV . The current per bond J (obtained in the staggered flux phase) is computed bytreating both ∆ t and ∆ φ as variational parameters. Thecomputed values are shown in Fig. 3a with labeling onthe right. One sees an opposite behaviour of the currentas a function of U with respect to Ω(∆ t, ∆ φ ) − Ω(∆ t ).Fig. 3b shows the values of variational parameters∆ φ sad and ∆ t sad at the saddle points as a function ofinteraction U at half filling. One can see that the ∆ φ sad increases as U is increasing in the small U region. Af-ter reaching its maximum around U ≈ .
5, where boththe difference Ω(∆ t, ∆ φ ) − Ω(∆ t ) and the current J be-come zero, ∆ φ sad drops sharply to zero. Note that, inthe regions of U < . U ≈ .
5, the difference be-tween Ω(∆ t, ∆ φ ) and Ω(∆ t ) is very small, in the sameorder as the accuracy of our calculation (1 . × − ). Asin Fig. 3b, the absolute value of variational parameter∆ t sad is decreasing as U increases. The slope of ∆ t sad curve changes abruptly at the place where the value of∆ φ sad becomes zero. B. Staggered flux phase in the Hubbard modelaway from half filling
In this section, we study the stability of staggered fluxphase when the system is away from half filling. We con-sider the competition between the staggered flux phase,superconductivity, and anti-ferromagnetism. Away fromhalf-filling, the shift in the chemical potential ∆ µ must betreated as variational parameter as well in order to satisfythe condition of thermodynamic consistency [33]. The d-wave Superconducting phases [30, 32] can be describedby introducing the corresponding Weiss field H ′ SC : H ′ SC = h SC X ij ∆ ij c i ↑ c j ↓ + H.c. ) , (6)where h SC is the strengths of the nearest-neighbor d-wave pairing field and ∆ ij is the d-wave form factor.Let us start by comparing the superconducting andstaggered flux states. In Fig. 4 we compare the grandpotential of the superconducting phase Ω(∆ µ, ∆ t, h SC )with that of the staggered flux phase Ω(∆ µ, ∆ t, ∆ φ ) fordifferent values of the interaction parameter U . One cansee that the values of the grand potential in the supercon-ducting state Ω(∆ µ, ∆ t, h SC ) and in the staggered flux Ω ( ∆ t , ∆ φ ) − Ω ( ∆ t ) U (a) J Ω(∆ t, ∆ φ ) − Ω(∆ t ) J ∆ φ s a d ( π ) U (b) −0.6−0.4−0.20.0 ∆ t s a d ∆ φ sad ∆ t sad FIG. 3: (Color online). (a) The grand potential differencebetween Ω(∆ t, ∆ φ ) and Ω(∆ t ) (denoted by (cid:3) with labelingon the left) and the current J corresponding to Ω(∆ t, ∆ φ )(denoted by △ with labeling on the right) as a function ofinteraction U . (b) The value of variational parameters at thesaddle points of Ω(∆ t, ∆ φ ), ∆ φ sad ( ∗ with labeling on the leftand in unit of π ) and ∆ t sad ( ▽ with labeling on the right), asa function of interaction U . The calculations are performedat half filling. phase Ω(∆ µ, ∆ t, ∆ φ ) are equal at half filling. This can beeasily understood because the d-wave pairing field H ′ SC in Eq. (6) is connected to the staggered-flux field H ′ T R inEq. (3) by a particle-hole transformation at half-filling[43, 44]. When away from half-filling, Ω(∆ µ, ∆ t, h SC ) isalways smaller than Ω(∆ µ, ∆ t, ∆ φ ) for all interactions,implying that the superconducting state is more stablethan the staggered flux phase in this region. For thelarge U case ( U & . φ is always zero as seen in Fig. 4(a)for U = 4 .
6. Fig. 4(a) shows that the values of thegrand potential Ω(∆ µ, ∆ t, h SC ) are smaller than that ofΩ(∆ µ, ∆ t ) and the difference between them is increasingas the chemical potential µ decreases. For smaller valuesof U a metastable VCA solution for staggered flux phasecan be obtained, that is, Ω(∆ µ, ∆ t, ∆ φ ) is smaller thanΩ(∆ µ, ∆ t ) with a finite value of variational parameter∆ φ . For the intermediate U in Fig. 4(b), the staggeredflux phase disappears at some value of µ . Finally, forsmall U the metastable VCA solution for the staggeredflux phase always exists for the whole range of chemical ∆ Ω ( a ) U = 4 . // Ω(∆ µ, ∆ t, ∆ φ ) − Ω(∆ µ, ∆ t )Ω(∆ µ, ∆ t, h SC ) − Ω(∆ µ, ∆ t ) ∆ Ω ( b ) U = 3 . µ ∆ Ω ( c ) U = 2 . FIG. 4: (Color online). The grand potential of super-conductor phase Ω(∆ µ, ∆ t, h SC ) and staggered-flux phaseΩ(∆ µ, ∆ t, ∆ φ ), with respect to Ω(∆ µ, ∆ t ), as a function ofchemical potential µ for various interactions U . ∆ µ , ∆ t , ∆ φ ,and h SC are variational parameters . potential as seen in Fig. 4(c). Note that the plots in Fig.4 are shown as a function of chemical potential, whichcorresponds to a doping in the range of 0 ∼ J (a) and the value of varia-tional phase factor ∆ φ sad (b) at the saddle points of thestaggered flux phase are plotted as a function of dop-ing δ for various interactions U . One can see that both J and ∆ φ sad are zero for large U & .
0, indicating noVCA solution for staggered flux phase in this region. Forintermediate U , J and ∆ φ sad decrease as the doping δ is increasing and drop to zero at some value of doping(e.g., δ = 0 .
05 for U = 3 . U (e.g., U = 2 . φ sad keeps as a finite value (around0.8 π ) and does not drop to zero. However, the staggeredflux phase is less stable than the superconducting phase,as shown in Fig. 4.Including the antiferromagnetic phase, we choose tolook at the results for the values of U = 3 .
2, wherethe difference between Ω(∆ t, ∆ φ ) and Ω(∆ t ) is max-imal at half filling (see Fig. 3a). In Fig. 6 weshow the grand potential Ω with different variationalparameters as a function of chemical potential µ . Itis clear that Ω(∆ µ, ∆ t, h AF ) is much smaller than bothΩ(∆ µ, ∆ t, h SC ) and Ω(∆ µ, ∆ t, ∆ φ ). Thus, the antiferro- J (a) U =4.6 U =3.4 U =3.2 U =3.0 U =2.0 δ ∆ φ s a d ( π ) (b) FIG. 5: (Color online). Current J (a) and the value ofvariational phase factor ∆ φ sad (b) at the saddle points ofΩ(∆ µ, ∆ t, ∆ φ ) as a function of doping δ for various interac-tions U . µ Ω Ω(∆ µ, ∆ t, ∆ φ )Ω(∆ µ, ∆ t )Ω(∆ µ, ∆ t,h SC )Ω(∆ µ, ∆ t,h AF ) FIG. 6: (Color online). Grand potential Ω with differentvariational parameters (∆ µ , ∆ φ , h SC , and h AF ) as a functionof the chemical potential µ for U = 3 . magnetic phase is the most stable state in this region ofchemical potential. For even smaller chemical potential(or larger doping), the antiferromagnetic phase becomesunstable towards the superconducting phase.In summary, we find that the staggered flux phase isless stable than the superconducting state for all interac-tions and chemical potentials considered. For values ofdoping not very far away from half filling, the antiferro-magentic phase would be the most stable phase. C. Staggered flux phase in the Hubbard modelwith nearest-neighbor interaction
In this section, we discuss the effect of nearest-neighbour interaction on the staggered-flux phase in theHubbard model. The nearest-neighbour interaction hasthe form: V P h ij i n i n j , where h ij i denotes the nearest-neghbour pairs. This interaction can be added to theHamiltonian of Eq. (1). In the following, we chooseto look at the point with a fixed on-site interaction U = 3 .
2. In Fig. 7(a), the grand potential difference∆Ω = Ω(∆ t, ∆ φ ) − Ω(∆ t ) is plotted as a function ofnearest-neighbour interaction V at half-filling. For a pos-itive V , ∆Ω is increasing to zero as V increases, imply-ing that a positive repulsive V disfavors the staggered-flux phase. However, adding a negative V will decreasethe value of ∆Ω. This means that an attractive nearest-neighbour interaction V is favorable for the formation ofthe staggered-flux phase. However, for negative V if | V | is large enough (e.g. V < − . −0.5 −0.25 0.0 0.25 0.5 0.75 1.0−0.006−0.004−0.0020.0 V ∆ Ω = Ω ( ∆ t , ∆ φ ) − Ω ( ∆ t ) (a) U =3.2 µ ∆ Ω = Ω ( ∆ µ , ∆ t , ∆ φ ) − Ω ( ∆ µ , ∆ t ) V =0.5(b) −0.6 −0.5 −0.4 −0.3 −0.2−0.008−0.007−0.006 µ V =-0.5(c) FIG. 7: (Color online). (a) The grand potential differ-ence Ω(∆ t, ∆ φ ) − Ω(∆ t ) as a function of nearest-neighborinteraction V at half-filling for a fixed on-site interaction U = 3 .
2. (b) and (c) show the grand potential differenceΩ(∆ µ, ∆ t, ∆ φ ) − Ω(∆ µ, ∆ t ) as a function of chemical poten-tial µ for nearest-neighbor interaction V = 0 . V = − . U = 3 . The effect of nearest-neighbour interaction V on sys-tem away from half-filling is shown in Fig. 7(b) and(c) for V = 0 . V = − . µ, ∆ t, ∆ φ ) − Ω(∆ µ, ∆ t ) is plotted as a func-tion of chemical potential µ . For positive V = 0 .
5, ∆Ωincreases slightly when µ deviates from the half-filling point µ = 3 . . < µ < . µ < . µ > . V = − .
5, ∆Ωremains nearly constant in the Mott gap region (i.e, − . < µ < − . µ < − .
425 or µ > − . µ, ∆ t, h AF ) − Ω(∆ µ, ∆ t ),is − . − . D. Staggered flux phase in the three bandHubbard model
Our work can be naturally extended to consider thethree band Hubbard model to search for the existence ofthe spontaneous time-reversal symmetry breaking states.Such states are still under debate and have probablybeen observed in true materials [22, 45]. In the proto-type realsitic systems, the high-Tc cuprates, a crucialrole is played by the CuO planes, in which the oxy-gen orbitals are explicit degrees of freedom. The elec-tron dynamics in CuO plane can be described by aminimal model, the three band Hubbard model, contain-ing the copper d x − y orbital and oxygen p x and p y or-bitals. Experiments show evidence for time-reversal sym-metry breaking in BSCCO with photoemmision [45] andin HgBa CuO δ with polarized neutron diffraction [22].Early theoretical works discussed the flux phase in thesingle band model with current order and proposed itin conection to the pseudo-gap state in high-Tc cuprates[1, 2, 11]. In order to stabilize the flux phase, Varmashows in a mean-field approach for three band model thenecessity to go beyond on-site interactions for an explicittreatment of the nearest-neighbor interactions betweencopper and oxygen [14–17]. Orbital currents were alsoobtained by more accurate theoretical calculations formulti-band Hubbard models with or without the inclu-sion of oxygen orbitals [23, 46]. Within this subsectionwe show some preliminary results of VCA calculationsfor the time-reversal symmetry breaking phase in threeband Hubbard model.The three band Hubbard model under consideration isdescribed by the Hamiltonian: H = X σ t ijdp ( d † iσ p jσ + h.c. ) + X
5, ∆ = ǫ p − ǫ d = 3 . U d = 8 . U p = 3 .
0, and V dp = 0 .
5, where we have taken the Cu-Ohopping t ijdp as unit of the energies. ∆ φ ( π ) Ω ( ∆ µ , ∆ φ ) µ = − . µ = − . µ = − . FIG. 8: (Color online). Grand potential Ω(∆ µ, ∆ φ ) of thethree band Hubbard model as a function of the values of vari-ational parameter ∆ φ . The inset shows the phase configura-tion used in the calculation. In the inset of Fig. 8 we show the CuO plane inwhich a 2 × d x − y ) and two O ( p x , p y ) sites. The implementa-tion of time-reversal symmetry breaking within the threeband model follows the description given in Sec. II. Inthe present calculations we have used a reduced numberof variational parameters, namely the shift in the chem-ical potential ∆ µ as well as the shift ∆ φ of the phase on each hopping terms, the later having the form: H ′ T R = X σ t ijdp e i ∆ φ dpδ ( d † iσ p jσ + h.c. )+ X
The central aim of this paper was to extended theVCA method to study the time-reversal symmetry bro-ken phase. We proposed to use two variational parame-ters, the hopping ∆ t and phase factor ∆ φ , to study thestaggered flux phase in the 2D Hubbard model. Our cal-culation suggest a metastable staggered flux phase in theintermediate and small U regions. For the case of halffilling, the flux phase appears to be the most stable at U ≈ .
2. This implies that the staggered flux phase, orat least its fluctuations, are more likely to be observedin the small U than in large U region. We also studiedthe stability of the staggered flux phase by comparingits grand potential with those of the superconductor andantiferromagnetic phases. We found that one of theselatter phases is always more stable than the staggeredflux phase both at as well as away from half filling.From our results on the one-band 2D Hubabrd modelwe cannot draw a clear picture of which interactions arecrucial to cause the orbital currents, although we clearlyobserve that a nearest-neighbour repulsion ( V >
0) de-creases the current, that would be present in the absenceof the intersite interaction. In contrast, an attractivenearest-neighbour interaction (
V <
0) leads to an in-crease in the current comparing with the V = 0 case.The staggered-flux order that emerges in the presenceof V < U . At last, we also extented the method for time-reversal symmetry breaking phase to study the three band Hubbard model. By using only the shifts of phaseson each hopping terms as variational parameter, we didnot find a stable VCA saddle point for the staggered fluxphase. Acknowledgments
L.C. acknowledges discussions with P. Chakrabortyand J. Kunes. X.L. thanks H. Allmaier, Z. B. Huang,and S. P. Kou for helpful discussions. This work wassupported by the National Natural Science Foundationof China (No. 11004164 and No. 10974163) and by theAustrian Science Fund (FWF) P18551-N16. [1] I. Affleck and J. B. Marston, Phys. Rev. B , 3774(1988).[2] J. B. Marston and I. Affleck, Phys. Rev. B , 11538(1989).[3] B. I. Halperin and T. M. Rice, Solid State Phys. , 115(1968).[4] G. Kotliar and J. Liu, Phys. Rev. B , 5142 (1988).[5] P. Lederer, D. Poilblanc, and T. M. Rice, Phys. Rev.Lett. , 1519 (1989).[6] F. C. Zhang, Phys. Rev. Lett. , 974 (1990).[7] D. J. Scalapino, S. R. White, and I. Affleck, Phys. Rev.B , 100506 (2001).[8] A. Macridin, M. Jarrell, and T. Maier, Phys. Rev. B ,113105 (2004).[9] P. W. Leung, Phys. Rev. B , 6112 (2000).[10] T. D. Stanescu and P. Phillips, Phys. Rev. B , 220509(2001).[11] S. Chakravarty, R. B. Laughlin, D. K. Morr, andC. Nayak, Phys. Rev. B , 094503 (2001).[12] C. Nayak, Phys. Rev. B , 4880 (2000).[13] H. J. Schulz, Phys. Rev. B , 2940 (1989).[14] C. M. Varma, Phys. Rev. B , 14554 (1997).[15] C. M. Varma, Phys. Rev. Lett. , 3538 (1999).[16] M. E. Simon and C. M. Varma, Phys. Rev. Lett. ,247003 (2002).[17] C. M. Varma, Phys. Rev. B , 155113 (2006).[18] M. Greiter and R. Thomale, Phys. Rev. Lett. , 027005(2007).[19] R. Thomale and M. Greiter, Phys. Rev. B , 094511(2008).[20] B. Fauque, Y. Sidis, V. Hinkov, S. Pailhes, C. T. Lin,X. Chaud, and P. Bourges, Phys. Rev. Lett. , 197001(2006).[21] G. J. MacDougall, A. A. Aczel, J. P. Carlo, T. Ito, J. Ro-driguez, P. L. Russo, Y. J. Uemura, S. Wakimoto, andG. M. Luke, Phys. Rev. Lett. , 017001 (2008).[22] Y. Li, V. Bal´edent, N. Barisic, Y. Cho, B. Fauqu´e,Y. Sidis, G. Yu, X. Zhao, P. Bourges, and M. Greven,Nature (2008).[23] C. Weber, A. L¨auchli, F. Mila, and T. Giamarchi, Phys.Rev. Lett. , 017005 (2009).[24] J. B. Marston, J. O. Fjærestad, and A. Sudbø, Phys. Rev. Lett. , 056404 (2002).[25] U. Schollw¨ock, S. Chakravarty, J. O. Fjærestad, J. B.Marston, and M. Troyer, Phys. Rev. Lett. , 186401(2003).[26] H. J. Schulz, Phys. Rev. B , R2959 (1996).[27] E. Orignac and T. Giamarchi, Phys. Rev. B , 7167(1997).[28] J. O. Fjærestad and J. B. Marston, Phys. Rev. B ,125106 (2002).[29] C. Wu, W. V. Liu, and E. Fradkin, Phys. Rev. B ,115104 (2003).[30] C. Dahnken, M. Aichhorn, W. Hanke, E. Arrigoni, andM. Potthoff, Phys. Rev. B , 245110 (2004).[31] S. Brehm, E. Arrigoni, M. Aichhorn, and W. Hanke, Eu-rophys. Lett. , 27005 (2010).[32] D. S´en´echal, P. L. Lavertu, M. A. Marois, and A. M. S.Tremblay, Phys. Rev. Lett. , 156404 (2005).[33] M. Aichhorn and E. Arrigoni, Europhys. Lett. , 117(2005).[34] M. Potthoff, M. Aichhorn, and C. Dahnken, Phys. Rev.Lett. , 206402 (2003).[35] C. Gros and R. Valenti, Phys. Rev. B , 418 (1993).[36] D. S´en´echal, D. Perez, and M. Pioro-Ladri´ere, Phys. Rev.Lett. , 522 (2000).[37] S. G. Ovchinnikov and I. S. Sandalov, Physica C ,607 (1989).[38] M. Potthoff, Eur. Phys. J. B , 429 (2003).[39] M. Potthoff, Eur. Phys. J. B , 335 (2003).[40] Due to the artificial symmetry breaking caused by thecluster partition, the order parameter is not uniform.Therefore, we obtain it as an average of the modulusof the current over all bonds inside the reference cluster.[41] D. R. Hofstadter, Phys. Rev. B , 2239 (1976).[42] M. Aichhorn, E. Arrigoni, M. Potthoff, and W. Hanke,Phys. Rev. B , 024508 (2006).[43] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. , 17 (2006).[44] I. Affleck, Z. Zou, T. Hsu, and P. W. Anderson, Phys.Rev. B , 745 (1988).[45] A. Kaminski, S. Rosenkranz, H. M. Fretwell, J. C. Cam-puzano, Z. Li, H. Raffy, W. G. Cullen, H. You, C. G.Olson, C. M. Varma, et al., Nature , 610 (2002). [46] P. Chudzinski, M. Gabay, and T. Giamarchi, Phys. Rev.B , 161101 (2007).[47] M. S. Hybertsen, M. Schl¨uter, and N. E. Christensen,Phys. Rev. B , 9028 (1989).[48] A. K. McMahan, J. F. Annett, and R. M. Martin, Phys. Rev. B , 6268 (1990).[49] G. Dopf, A. Muramatsu, and W. Hanke, Phys. Rev. Lett.68